The free field is constructed from an irreducible representation of the Poincaré group
through second quantization. We use the notation which is to some extent consistent with
[1]}. The one-particle Hilbert space is \({\mathcal {H}}_1 := L^2({\mathbb {R}}, d\theta )\)  and for
the mass \(m > 0\)  the (proper orthochronous) Poincaré group acts by
\(U_1(a,\lambda )\psi (\theta ) = e^{ip(\theta )\cdot a}\psi (\theta -\lambda )\) , where
\(p(\theta ) := (m\cosh (\theta ), m\sinh (\theta ))\)  parametrizes the mass shell.
We introduce the (auxiliary) unsymmetrized Hilbert space \({\mathcal {H}}^\Sigma := \bigoplus {\mathcal {H}}_1^{\otimes n}\)  and
the (physical) symmetrized Hilbert space \(\mathcal {H}_{\mathrm {r}}:= \bigoplus Q_n {\mathcal {H}}_1^{\otimes n}\) , where \(Q_n\) 
is the projection onto the symmetric subspace.
